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Edinburgh Research Explorer Edinburgh Research Explorer Signatures in algebra, topology and dynamics Citation for published version: Ghys, E & Ranicki, A 2016, 'Signatures in algebra, topology and dynamics' Ensaios Matemáticos, vol. 30, pp. 1-173. Link: Link to publication record in Edinburgh Research Explorer Document Version: Peer reviewed version Published In: Ensaios Matemáticos General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 05. Apr. 2019 Signatures in algebra, topology and dynamics Etienne´ Ghys, Andrew Ranicki 31st December, 2015 arXiv:1512.09258v1 [math.AT] 31 Dec 2015 E.G A.R. UMPA UMR 5669 CNRS School of Mathematics, ENS Lyon University of Edinburgh Site Monod James Clerk Maxwell Building 46 All´eed'Italie Peter Guthrie Tait Road 69364 Lyon Edinburgh EH9 3FD France Scotland, UK [email protected] [email protected] 1 Contents 1 Algebra 6 1.1 The basic definitions . .6 1.2 Linear congruence . .7 1.3 The signature of a symmetric matrix . 11 1.4 Orthogonal congruence . 13 1.5 Continued fractions, tridiagonal matrices and signatures . 22 1.6 Sturm's theorem and its reformulation by Sylvester . 27 1.7 Witt groups of fields . 36 1.8 The Witt groups of the function field R(X), ordered fields and Q.................................. 42 2 Topology 53 2.1 Even-dimensional manifolds . 53 2.2 Cobordism . 58 2.3 Odd-dimensional manifolds . 61 2.4 The union of manifolds with boundary; Novikov additivity and Wall nonadditivity of the signature . 65 2.5 Plumbing: from quadratic forms to manifolds . 72 3 Knots, links, braids and signatures 78 3.1 Knots, links, Seifert surfaces and complements . 78 3.2 Signatures of knots : a tale from the sixties . 83 3.3 Two great papers by Milnor and the modern approach . 84 3.4 Braids . 89 3.5 Knots and signatures in higher dimensions . 93 4 The symplectic group and the Maslov class 95 4.1 A very brief history of the symplectic group . 95 4.2 The Maslov class . 98 4.3 A short recollection of group cohomology . 100 4.4 The topology of Λn ........................ 101 4.5 Maslov indices and the universal cover of Λn .......... 102 4.6 The Maslov class as a bounded cohomology class . 105 4.7 Picture in the case n =2..................... 109 2 5 Dynamics 111 5.1 The Calabi and Ruelle invariant . 111 5.2 The Conley-Zehnder index . 112 5.3 Knots and dynamics . 114 6 Number theory, topology and signatures 117 6.1 The modular group SL(2; Z)................... 117 6.2 Torus bundles and their signatures . 119 6.3 Lens spaces . 123 6.4 Dedekind sums . 125 7 Appendix: Algebraic L-theory of rings with involution and the localization exact sequence 128 7.1 Forms over a ring with involution . 129 7.2 The localization exact sequence . 133 3 Introduction There is no doubt that quadratic forms are among the most basic and impor- tant objects in mathematics. The distinction between an ellipse, a parabola and a hyperbola belongs to the standard curriculum of undergraduate stu- dents in mathematics, and Sylvester's 1852 Law of Inertia for quadratic forms represents the formalization of this fact. In a suitable basis, any quadratic form in Rn can be written as 2 2 2 2 2 x1 + x2 + ··· + xp − xp+1 − · · · − xp+q for some integers 0 6 p; q 6 n with p + q 6 n. These integers are uniquely defined by the quadratic form, and are the same for two forms if and only if the forms are linearly congruent. The sum p+q is the rank and the difference p−q is the signature of the quadratic form. In other words, Sylvester's Law of Inertia asserts that for any n > 1 the number of linear congruence classes of n quadratic forms in R is the number of ordered pairs (p; q) with 0 6 p; q 6 n, p + q 6 n (which is equal to (n + 1)(n + 2)=2). It is not a surprise that from these humble beginnings the theory of quadratic forms has percolated to the full body of mathematics, in many different guises. It would require a large encyclopedia to keep track of all these appearances, and it is certainly not our goal to provide such a com- pendium here. In this survey paper, we would like to present some morceaux choisis with the main purpose of serving as a general introduction to the other papers in these volumes, which deal with the signatures of braids. As the title suggests, we shall concentrate on algebra, topology, and dynamics. Even though all the material that we describe is standard, it is usually scattered in the literature. For instance, books and papers dealing with Hamiltonian dynamics usually do not discuss Wall non additivity of signatures. Section 1 contains very standard facts about quadratic forms in Rn. We tried to give some historical flavour to our presentation. A common thread will be Sturm's theorem, counting real roots of polynomials. From its original formulation as a number of sign variations, we shall describe its reformulation by Sylvester as the signature of a symmetric matrix, and then as an equality in a suitable Witt group. Section 2 is topological. The homology of a manifold is equipped with intersection forms. Therefore signatures came into play, around 1940, and 4 serve as invariants of manifolds. Section 3 deals with the Maslov class. One could easily write thick books on this topic (that probably nobody would read) and we had to limit our- selves to the basics. It is amazing how ubiquitous this class can be and it is sometimes difficult to recognize it. There are connections with mathematical physics, topology, number theory, dynamics, geometry, bounded cohomology etc. In the final sections, we choose three typical applications, among many other possibilities: Section 4 presents some aspects of dynamical systems, specifically hamil- tonian, for which signatures are relevant. Section 5 goes deeper in some topological aspects. In particular, we de- scribe the link between the Wall non additivity and the Maslov class. Section 6 describes some fascinating connections with number theory. Section 7 is an appendix on the algebraic L-theory used in surgery ob- struction theory, which generalizes some of the algebraic techniques presented here to forms over (noncommutative) rings with involution. We thank the American Institute of Mathematics in Palo Alto, at which we organized a meeting on the \Many facets of Maslov invariants" in April 2014. Here are our signatures: 5 1 Algebra 1.1 The basic definitions Let us begin with some very basic definitions from linear algebra. In this section we shall be mainly considering matrices over R. An m × n matrix S = (sij) with entries sij 2 R (1 6 i 6 m, 1 6 j 6 n) corresponds to a bilinear form (or pairing) of real vector spaces: m n m n X X S : ((x1; x2; : : : ; xm); (y1; y2; : : : ; yn)) 2 R × R 7! sijxiyj 2 R: i=1 j=1 ∗ ∗ The transpose of an m×n matrix S = (sij) is the n×m matrix S = (sji) ∗ with sji = sij: An n × n matrix S is symmetric if S∗ = S. Quadratic forms correspond to symmetric matrices S. For v = (x1; : : : ; xn) 2 Rn, we set Q(v) = S(v; v); with S(v1; v2) = (Q(v1 + v2) − Q(v1) − Q(v2))=2: A quadratic form Q(V ) (or the corresponding symmetric matrix S) is positive definite (resp. negative definite) if Q(v) > 0 (resp. Q(v) < 0) for all non zero v 2 Rn. Two symmetric n × n matrices S; T are linearly congruent if T = A∗SA for an invertible n × n matrix A. A linear congruence can be regarded as an isomorphism A : Rn ! Rn of real vector spaces such that T (v1; v2) = S(Av1; Av2) 2 R: The spectrum of an n × n matrix S is the set of eigenvalues, i.e. the roots λ 2 C of the characteristic polynomial det(XIn − S) 2 R[X]. In general, linearly congruent symmetric matrices S; T do not have the same characteristic polynomial and spectrum. However, it is significant (and by no means obvious) that symmetric matrices have real eigenvalues, and that the eigenvalues of linearly congruent symmetric matrices have the same signs. Two n × n matrices S; T are conjugate if T = A−1SA for an invertible n × n matrix A, in which case S; T have the same characteristic polynomial det(XIn − T ) = det(XIn − S) and hence the same eigenvalues. An n × n 6 matrix A is orthogonal if it is invertible and A−1 = A∗. This is equivalent to 2 2 n the preservation of the quadratic form x1 +···+xn in R . Two n×n matrices S; T are orthogonally congruent if T = A∗SA for an orthogonal n × n matrix A, in which case they are both linearly congruent and conjugate. 1.2 Linear congruence The first basic theorem is that any quadratic form Q(x1; : : : ; xn) with real coefficients can be written as a sum, or difference, of squares of linear forms.
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